Three-degrees-of-freedom orientation manipulation of small untethered robots with a single anisotropic soft magnet

Magnetic actuation has been well exploited for untethered manipulation and locomotion of small-scale robots in complex environments such as intracorporeal lumens. Most existing magnetic actuation systems employ a permanent magnet onboard the robot. However, only 2-DoF orientation of the permanent-magnet robot can be controlled since no torque can be generated about its axis of magnetic moment, which limits the dexterity of manipulation. Here, we propose a new magnetic actuation method using a single soft magnet with an anisotropic geometry (e.g., triaxial ellipsoids) for full 3-DoF orientation manipulation. The fundamental actuation principle of anisotropic magnetization and 3-DoF torque generation are analytically modeled and experimentally validated. The hierarchical orientation stability about three principal axes is investigated, based on which we propose and validate a multi-step open-loop control strategy to alternatingly manipulate the direction of the longest axis of the soft magnet and the rotation about it for dexterous 3-DoF orientation manipulation.

tank to suspend the capsule robot is extracted by a water pump and meanwhile pumped back to the tank.The water inlet (red arrow) is at the bottom of the tank and the water outlet (blue arrow) is near the surface.The water circulation causes a flow disturbance to the orientation manipulation of the capsule robot.The water flow rate is set to be 100mL/min and the volume of water in the tank is about 750mL.Supplementary Fig. 6 | 3-DoF orientation manipulation in the environmental flow field.The experiment of 3-DoF orientation manipulation in the environmental flow field is carried out using a water circulation system (Supplementary Fig. 5).The orientation path is the same as that in Fig. 6   Supplementary Table 1 | Experimental data for validation of maximum magnetic torques about three principal axes of the soft magnet.We shot the video of the soft magnet that rotates about three principal axes for 45° to align with the magnetic field and recorded the time of rotation to estimate the maximum torque.The experiments are conducted under different magnitudes of the applied field by tuning current in the coils.For each video, we calculate its frame rate dividing the total video frames by the total video duration.Then the effective rotation frames are counted.Finally, the rotation time is calculated according to the frame rate and the effective rotation frames.The method of estimating the maximum torque from rotation time is given by the section of "Method" in the main text.The moment of inertia of the capsule robot including the soft magnet about a-axis, baxis, c-axis are 1,255.849×10-9 kg⋅m 2 , 4,092.301×10 - kg⋅m 2 , 4,278.615×10 - kg⋅m 2 , respectively.

Demagnetization factors of an anisotropic triaxial ellipsoid
To compute demagnetization factors of an anisotropic ellipsoid, we employ the method provided in [1].Firstly, the following assumption is made: where a, b, and c are the semi-axial lengths of the ellipsoid's three principal axes.The corresponding demagnetization factors are   ,   and   .The equations to calculate   ,   and   are given below: where (, ), (, ) are incomplete elliptic integrals of the first and second kind,  is the modulus and  is the amplitude of these integrals.And   ,   and   follow the rule that:

Demagnetization factors of an anisotropic cuboid
The formulas for calculating the demagnetization factors of a cuboid are given below [2]: where 2, 2, and 2 represent the length, width, and height of the cuboid, respectively, and   denotes the demagnetization factor along the axis of height.By replacing  with ,  with , and  with  in Equation ( 9), the formula for calculating   is obtained.Similarly, by replacing  with ,  with , and  with  in Equation ( 9), the formula for calculating   is obtained.  ,   and   also follow Equation (8).

Demagnetization factors of an anisotropic elliptic cylinder
Consider an elliptic cylinder with semi-axial length  and , thickness (height) .Assume that

Supplementary Table 2 | Symbols and parameters characterizing the elliptic cylinder
The formulas for calculation of magnetization factors of elliptical cylinders are given by [3]: We use the explicit first six terms of   () and   () to calculate a good approximation value of   and   : where (  2 ) and (  2 ) are complete elliptic integrals of the first and the second kind.Finally,   can be calculated by the unity constraint: Supplementary Note 2: Preliminary results on modeling of 3-DoF magnetic force and 6-DoF manipulation of soft-magnet robots

Modeling of 3-DoF magnetic force on an anisotropic soft magnet
Magnetic force on a magnetic dipole  depends on the spatial gradient of the local magnetic field , which is given below Now replace the magnetic dipole with an anisotropic soft magnet, with a-, b-, and c-axis of the soft magnet aligned with the x-, y-, and z-axis of the world coordinate system (WCS) as shown in Supplementary Fig. 7 (a).The soft magnet's magnetization is determined by Equation ( 3) -(7) in the paper.Combining these equations with Equation (15), we can obtain magnetic force  on the soft magnet in an external magnetic field : where  is the volume of the soft magnet and  0 is the permeability in vacuum.From Equation ( 16), to apply a desired force to the soft magnet, we need to provide both an appropriate magnetic field and an appropriate magnetic field spatial gradient.Equation ( 16) also shows that it is feasible to generate magnetic forces on the soft magnet in three directions for 3-DoF translation control.

Supplementary
As shown in Supplementary Fig. 7 (a) and (b), when  is applied in an arbitrary direction without a magnetic field spatial gradient, the soft magnet experiences a torque that causes its a-axis to align with the direction of the magnetic field according to the analysis of orientation stability in the paper.
Once aligned, the magnetic moment  of the soft magnet under  is given by: Substitute Equation ( 18) into (17), the following equation of magnetic force is obtained: shown in Supplementary Fig. 7 (c).

A preliminary control method for 6-DoF manipulation
In the following section, we will introduce a control strategy that we are currently Similarly, we can control the magnetic force in y-axis and z-axis by tuning (b) in the paper.The same control accuracy is achieved as the case without flow disturbance, which shows the robustness of the proposed soft-magnet manipulation method.The scale bars indicate 2 cm.

≤ 1 .
≥  and define the ratio  =     ,   and   are demagnetization factors along three dimensions of the elliptic cylinder.The parameters employed in calculation of demagnetization factors for elliptic cylinders are summarized in Supplementary

Fig. 7 |B
Magnetic force on soft magnet in a magnetic with spatial gradient.a. Schematic of anisotropic magnetization and magnetic torque of a tri-ellipsoidal soft magnet.The anisotropic magnetization of the soft magnet leads to separation of the direction of magnetization vector  from the applied field , resulting in a non-zero magnetic torque .b.Alignment of an ellipsoidal soft magnet's a-axis with the direction of magnetic field  due to the torque  .c. Schematic of 3-DoF magnetic force on the soft magnet subjected to both magnetic field  and magnetic field gradient ∇.The orthogonal Helmholtz coils used in the paper can only generate a uniform magnetic field but lacks the capability to produce magnetic field spatial gradients.Therefore, we need to design a new magnetic field generation setup, i.e., the Helmholtz-Maxwell combination coils.The Helmholtz-Maxwell coils consist of three pairs of Helmholtz coils and three pairs of Maxwell coils, both of which are installed orthogonally.A uniform magnetic field can be generated in any direction by Helmholtz coils that runs the currents in the same direction in pairs of coils, while three magnetic field gradients, namely provided by Maxwell coils that runs the currents in the opposite direction in pairs of coils.In the working space of the setup, the magnitude of By analyzing Equation (19), we can draw the following conclusion: Given that a-axis of the soft magnet aligns with the direction of , we can control the magnetic force  on the soft magnet by regulating the terms   -Maxwell combination coils, as

conceptualizing for 6 -
DoF manipulation of the soft-magnet robot.This 6-DoF control strategy is built upon the open-loop 3-DoF orientation control method proposed in this paper.Supplementary Fig. 8 (a), (b), and (c) illustrate translation control in three directions.Taking translation control in x-axis as an example, a magnetic field  is applied in x-axis to align the a-axis of the soft magnet and magnetic moment  with the field by activating the Helmholtz coils along the x-axis.Then activate the Maxwell coils along the x-axis to generate    while maintaining  .According to Equation (19), the force can be expressed as  = control the magnetic force in x-axis by tuning only    .